EDP Sciences logo
Subscriber Authentication Point
Sciengine logo
Search
Menu
All issues Volume 60 / No 4 (July-August 2026) ESAIM: M2AN, 60 4 (2026) 1575-1599 References
Open Access
Issue
ESAIM: M2AN
Volume 60, Number 4, July-August 2026
Page(s) 1575 - 1599
DOI https://doi.org/10.1051/m2an/2026040
Published online 08 July 2026
  1. B. Aksoylu, On four mutual properties of classical and nonlocal wave equations. J. Peridyn. Nonlocal Model. 5 (2023) 60–80. [Google Scholar]
  2. B. Aksoylu and T. Mengesha, Results on nonlocal boundary value problems. Numer. Funct. Anal. Optim. 31 (2010) 1301–1317. [Google Scholar]
  3. B. Alali and N. Albin, Fourier spectral methods for nonlocal models. J. Peridyn. Nonlocal Model. 2 (2020) 317–335. [Google Scholar]
  4. B. Alali and N. Albin, Fourier multipliers for nonlocal laplace operators. Appl. Anal. 100 (2021) 2526–2546. [Google Scholar]
  5. P.W. Bates and A. Chmaj, An integrodifferential model for phase transitions: stationary solutions in higher space dimensions. J. Stat. Phys. 95 (1999) 1119–1139. [CrossRef] [Google Scholar]
  6. H.R. Beyer, B. Aksoylu and F. Celiker, On a class of nonlocal wave equations from applications. J. Math. Phys. 57 (2016) 062902. [Google Scholar]
  7. F. Bobaru and M. Duangpanya, The peridynamic formulation for transient heat conduction. Int. J. Heat Mass Transf. 53 (2010) 4047–4059. [Google Scholar]
  8. A. Buades, B. Coll and J.-M. Morel, Image denoising methods. A new nonlocal principle. SIAM Rev. 52 (2010) 113–147. [Google Scholar]
  9. F.A. Chiarello and P. Goatin, Global entropy weak solutions for general non-local traffic flow models with anisotropic kernel. ESAIM: M2AN 52 (2018) 163–180. [CrossRef] [EDP Sciences] [Google Scholar]
  10. C.-S. Chou, C.-W. Shu and Y. Xing, Optimal energy conserving local discontinuous Galerkin methods for second-order wave equation in heterogeneous media. J. Comput. Phys. 272 (2014) 88–107. [Google Scholar]
  11. P.G. Ciarlet, The Finite Element Method for Elliptic Problems. SIAM (2002). [Google Scholar]
  12. B. Cockburn, Discontinuous Galerkin methods for computational fluid dynamics, Encyclopedia of Computational Mechanics 3 (2004) 91–127. [Google Scholar]
  13. B. Cockburn, J. Gopalakrishnan and R. Lazarov, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems. SIAM J. Numer. Anal. 47 (2009) 1319–1365. [CrossRef] [MathSciNet] [Google Scholar]
  14. G.M. Coclite, A. Fanizzi, L. Lopez, F. Maddalena and S.F. Pellegrino, Numerical methods for the nonlocal wave equation of the peridynamics. Appl. Numer. Math. 155 (2020) 119–139. [Google Scholar]
  15. T. Dang, B. Alali and N. Albin, Regularity of solutions for the nonlocal wave equation on periodic distributions. Preprint arXiv:2408.00912 (2024). [Google Scholar]
  16. K. Dayal and K. Bhattacharya, A real-space non-local phase-field model of ferroelectric domain patterns in complex geometries. Acta Mater. 55 (2007) 1907–1917. [Google Scholar]
  17. Q. Du, M. Gunzburger, R.B. Lehoucq and K. Zhou, Analysis and approximation of nonlocal diffusion problems with volume constraints. SIAM Rev. 54 (2012) 667–696. [CrossRef] [MathSciNet] [Google Scholar]
  18. Q. Du, H. Han, J. Zhang and C. Zheng, Numerical solution of a two-dimensional nonlocal wave equation on unbounded domains. SIAM J. Sci. Comput. 40 (2018) A1430–A1445. [CrossRef] [Google Scholar]
  19. Q. Du, L. Ju and J. Lu, A discontinuous Galerkin method for one-dimensional time-dependent nonlocal diffusion problems. Math. Comput. 88 (2019) 123–147. [Google Scholar]
  20. Q. Du, L. Ju, J. Lu and X. Tian, A discontinuous Galerkin method with penalty for one-dimensional nonlocal diffusion problems. Commun. Appl. Math. Comput. 2 (2020) 31–55. [CrossRef] [MathSciNet] [Google Scholar]
  21. Q. Du, L. Ju, J. Lu and X. Tian, Numerical analysis of a class of penalty discontinuous Galerkin methods for nonlocal diffusion problems. ESAIM: M2AN 58 (2024) 2035–2059. [Google Scholar]
  22. Q. Du, Y. Tao, X. Tian and J. Yang, Asymptotically compatible discretization of multidimensional nonlocal diffusion models and approximation of nonlocal Green’s functions. IMA J. Numer. Anal. 39 (2019) 607–625. [CrossRef] [MathSciNet] [Google Scholar]
  23. Y. Du and J. Zhang, On perfectly matched layers of nonlocal wave equations in unbounded multiscale media. J. Mech. Mater. Struct. 17 (2023) 343–364. [Google Scholar]
  24. Y. Du, L. Zhang and Z. Zhang, Convergence analysis of a discontinuous Galerkin method for wave equations in second-order form. SIAM J. Numer. Anal. 57 (2019) 238–265. [Google Scholar]
  25. M. D’Elia, Q. Du, C. Glusa, M. Gunzburger, X. Tian and Z. Zhou, Numerical methods for nonlocal and fractional models. Acta Numer. 29 (2020) 1–124. [CrossRef] [MathSciNet] [Google Scholar]
  26. E. Emmrich and O. Weckner, Analysis and numerical approximation of an integro-differential equation modeling non-local effects in linear elasticity. Math. Mech. Solids 12 (2007) 363–384. [Google Scholar]
  27. E. Emmrich and O. Weckner, On the well-posedness of the linear peridynamic model and its convergence towards the Navier equation of linear elasticity. Commun. Math. Sci. 5 (2007) 851–864. [Google Scholar]
  28. P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, in Trends in Nonlinear Analysis, Edited by M. Kirkilionis, S. Krömker, R. Rannacher, F. Tomi. 1st edition. Berlin, Heidelberg (2003) 153–191. [Google Scholar]
  29. M. Foss and P. Radu, Differentiability and integrability properties for solutions to nonlocal equations, in New Trends in Differential Equations, Control Theory and Optimization: Proceedings of the 8th Congress of Romanian Mathematicians. World Scientific (2016) 105–119. [Google Scholar]
  30. G. Gilboa and S. Osher, Nonlocal operators with applications to image processing. Multiscale Model. Simul. 7 (2009) 1005–1028. [Google Scholar]
  31. M.J. Grote, A. Schneebeli and D. Schötzau, Discontinuous Galerkin finite element method for the wave equation. SIAM J. Numer. Anal. 44 (2006) 2408–2431. [Google Scholar]
  32. Q. Guan and M. Gunzburger, Stability and accuracy of time-stepping schemes and dispersion relations for a nonlocal wave equation. Numer. Methods Partial Differ. Equ. 31 (2015) 500–516. [Google Scholar]
  33. A. Hermann, A. Shojaei, P. Seleson, C.J. Cyron and S.A. Silling, Dirichlet-type absorbing boundary conditions for peridynamic scalar waves in two-dimensional viscous media. Int. J. Numer. Methods Eng. 124 (2023) 3524–3553. [Google Scholar]
  34. N.C. Nguyen, J. Peraire and B. Cockburn, High-order implicit hybridizable discontinuous Galerkin methods for acoustics and elastodynamics. J. Comput. Phys. 230 (2011) 3695–3718. [Google Scholar]
  35. W. Reed and T. Hill, Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Scientific Laboratory report LA-UR-73-479 (1973). [Google Scholar]
  36. K. Ren, L. Zhang and Y. Zhou, An energy-based discontinuous Galerkin method for the nonlinear Schrödinger equation with wave operator. SIAM J. Numer. Anal. 62 (2024) 2459–2483. [Google Scholar]
  37. L. Rosasco, M. Belkin and E. De Vito, On learning with integral operators. J. Mach. Learn. Res. 11 (2010) 905–934. [Google Scholar]
  38. E.W. Sachs and M. Schu, A priori error estimates for reduced order models in finance. ESAIM: M2AN 47 (2013) 449–469. [Google Scholar]
  39. S.A. Silling, Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids 48 (2000) 175–209. [CrossRef] [MathSciNet] [Google Scholar]
  40. H. Tian, L. Ju and Q. Du, A conservative nonlocal convection–diffusion model and asymptotically compatible finite difference discretization. Comput. Methods Appl. Mech. Eng. 320 (2017) 46–67. [Google Scholar]
  41. X. Tian and Q. Du, Analysis and comparison of different approximations to nonlocal diffusion and linear peridynamic equations. SIAM J. Numer. Anal. 51 (2013) 3458–3482. [Google Scholar]
  42. X. Tian and Q. Du, Asymptotically compatible schemes and applications to robust discretization of nonlocal models. SIAM J. Numer. Anal. 52 (2014) 1641–1665. [Google Scholar]
  43. X. Tian and Q. Du, Nonconforming discontinuous Galerkin methods for nonlocal variational problems. SIAM J. Numer. Anal. 53 (2015) 762–781. [CrossRef] [MathSciNet] [Google Scholar]
  44. S.V.N. Vishwanathan, N.N. Schraudolph, R. Kondor and K.M. Borgwardt, Graph kernels. J. Mach. Learn. Res. 11 (2010) 1201–1242. [Google Scholar]
  45. J. Wang, J.Z. Yang and J. Zhang, Stability and convergence analysis of high-order numerical schemes with DtN-type absorbing boundary conditions for nonlocal wave equations. Preprint arXiv:2211.04307 (2022). [Google Scholar]
  46. L. Wang, J. Xu and J. Wang, A peridynamic framework and simulation of non-Fourier and nonlocal heat conduction. Int. J. Heat Mass Transf. 118 (2018) 1284–1292. [Google Scholar]
  47. O. Weckner and R. Abeyaratne, The effect of long-range forces on the dynamics of a bar. J. Mech. Phys. Solids 53 (2005) 705–728. [CrossRef] [MathSciNet] [Google Scholar]
  48. H. You, X. Lu, N. Task and Y. Yu, An asymptotically compatible approach for Neumann-type boundary condition on nonlocal problems. ESAIM: M2AN 54 (2020) 1373–1413. [Google Scholar]
  49. C. Zheng, J. Hu, Q. Du and J. Zhang, Numerical solution of the nonlocal diffusion equation on the real line. SIAM J. Sci. Comput. 39 (2017) A1951–A1968. [CrossRef] [Google Scholar]

Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.

Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.

Initial download of the metrics may take a while.

Recommended for you